If an infinite resistive network, whose edges have resistance 1 ohm, satisfies a certain graph theoretical condition, then the homogeneous Kirchhoff equations have no nonzero solutions vanishing at infinity. Every vertex transitive graph with polynomial growth satisfies such a condition. Furthermore uniqueness holds in Cartesian products of infinite regular graphs. Graphs with more than one end and satisfying an isoperimetric inequality provide a counterexample to uniqueness. These results extend partially also to networks with nonconstant resistances. © 1991.
If an infinite resistive network, whose edges have resistance 1 ohm, satisfies a certain graph theoretical condition, then the homogeneous Kirchhoff equations have no nonzero solutions vanishing at infinity. Every vertex transitive graph with polynomial growth satisfies such a condition. Furthermore uniqueness holds in Cartesian products of infinite regular graphs. Graphs with more than one end and staisfying an isoperimetric inequality provide a counterexample to uniqueness. These results extend partially also to networks with nonconstant resistances
Soardi, P., Woess, W. (1991). Uniqueness of currents in infinite resistive networks. DISCRETE APPLIED MATHEMATICS, 31(1), 37-49 [10.1016/0166-218X(91)90031-Q].
Uniqueness of currents in infinite resistive networks
SOARDI, PAOLO MAURIZIO;
1991
Abstract
If an infinite resistive network, whose edges have resistance 1 ohm, satisfies a certain graph theoretical condition, then the homogeneous Kirchhoff equations have no nonzero solutions vanishing at infinity. Every vertex transitive graph with polynomial growth satisfies such a condition. Furthermore uniqueness holds in Cartesian products of infinite regular graphs. Graphs with more than one end and satisfying an isoperimetric inequality provide a counterexample to uniqueness. These results extend partially also to networks with nonconstant resistances. © 1991.
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